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Journal Cover Physica D: Nonlinear Phenomena
  [SJR: 1.049]   [H-I: 102]   [3 followers]  Follow
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
   ISSN (Print) 0167-2789
   Published by Elsevier Homepage  [3043 journals]
  • Wave turbulence theory of elastic plates
    • Authors: Gustavo Düring; Christophe Josserand; Sergio Rica
      Pages: 42 - 73
      Abstract: Publication date: 15 May 2017
      Source:Physica D: Nonlinear Phenomena, Volume 347
      Author(s): Gustavo Düring, Christophe Josserand, Sergio Rica
      This article presents the complete study of the long-time evolution of random waves of a vibrating thin elastic plate in the limit of small plate deformation so that modes of oscillations interact weakly. According to the wave turbulence theory a nonlinear wave system evolves in longtime creating a slow redistribution of the spectral energy from one mode to another. We derive step by step, following the method of cumulants expansion and multiscale asymptotic perturbations, the kinetic equation for the second order cumulants as well as the second and fourth order renormalization of the dispersion relation of the waves. We characterize the non-equilibrium evolution to an equilibrium wave spectrum, which happens to be the well known Rayleigh–Jeans distribution. Moreover we show the existence of an energy cascade, often called the Kolmogorov–Zakharov spectrum, which happens to be not simply a power law, but a logarithmic correction to the Rayleigh–Jeans distribution. We perform numerical simulations confirming these scenarii, namely the equilibrium relaxation for closed systems and the existence of an energy cascade wave spectrum. Both show a good agreement between theoretical predictions and numerics. We show also some other relevant features of vibrating elastic plates, such as the existence of a self-similar wave action inverse cascade which happens to blow-up in finite time. We discuss the mechanism of the wave breakdown phenomena in elastic plates as well as the limit of strong turbulence which arises as the thickness of the plate vanishes. Finally, we discuss the role of dissipation and the connection with experiments, and the generalization of the wave turbulence theory to elastic shells.

      PubDate: 2017-03-28T02:42:26Z
      DOI: 10.1016/j.physd.2017.01.002
      Issue No: Vol. 347 (2017)
  • A numerical estimate of the regularity of a family of Strange Non-Chaotic
    • Authors: Lluís Alsedà i Soler; Josep Maria Mondelo González; David Romero i Sànchez
      Pages: 74 - 89
      Abstract: Publication date: 15 May 2017
      Source:Physica D: Nonlinear Phenomena, Volume 347
      Author(s): Lluís Alsedà i Soler, Josep Maria Mondelo González, David Romero i Sànchez
      We estimate numerically the regularities of a family of Strange Non-Chaotic Attractors related with one of the models studied in (Grebogi et al., 1984) (see also Keller, 1996). To estimate these regularities we use wavelet analysis in the spirit of de la Llave and Petrov (2002) together with some ad-hoc techniques that we develop to overcome the theoretical difficulties that arise in the application of the method to the particular family that we consider. These difficulties are mainly due to the facts that we do not have an explicit formula for the attractor and it is discontinuous almost everywhere for some values of the parameters. Concretely we propose an algorithm based on the Fast Wavelet Transform. Also a quality check of the wavelet coefficients and regularity estimates is done.

      PubDate: 2017-03-28T02:42:26Z
      DOI: 10.1016/j.physd.2016.12.006
      Issue No: Vol. 347 (2017)
  • Full analysis of small hypercycles with short-circuits in prebiotic
    • Authors: Josep Sardanyés; J. Tomás Lázaro; Antoni Guillamon; Ernest Fontich
      Pages: 90 - 108
      Abstract: Publication date: 15 May 2017
      Source:Physica D: Nonlinear Phenomena, Volume 347
      Author(s): Josep Sardanyés, J. Tomás Lázaro, Antoni Guillamon, Ernest Fontich
      It is known that hypercycles are sensitive to the so-called parasites and short-circuits. While the impact of parasites has been widely investigated for well-mixed and spatial hypercycles, the effect of short-circuits in hypercycles remains poorly understood. In this article we analyze the mean field and spatial dynamics of two small, asymmetric hypercycles with short-circuits. Specifically, we analyze a two-member hypercycle where one of the species contains an auto-catalytic loop, as the simplest hypercycle with a short-circuit. Then, we extend this system by adding another species that closes a three-member hypercycle while keeping the auto-catalytic short-circuit and the two-member cycle. The mean field model allows us to discard the presence of stable or unstable periodic orbits for both systems. We characterize the bifurcations and transitions involved in the dominance of the short-circuits i.e., in the reduction of the hypercycles’ size. The spatial simulations reveal a random-like and mixed distribution of the replicators in the all-species coexistence, ruling out the presence of large-scale spatial patterns such as spirals or spots typical of larger, oscillating hypercycles. A Monte Carlo sampling of the parameter space for the well-mixed and the spatial models reveals that the probability of finding stable hypercycles with short-circuits drastically diminishes from the two-member to the three-member system, especially at growing degradation rates of the replicators. These findings pose a big constraint in the increase of hypercycle’s size and complexity under the presence of inner cycles, suggesting the importance of a rapid growth of hypercycles able to generate spatial structures (e.g., rotating spirals) prior to the emergence of inner cycles. Our results can also be useful for the future design and implementation of synthetic cooperative systems containing catalytic short-circuits.

      PubDate: 2017-03-28T02:42:26Z
      DOI: 10.1016/j.physd.2016.12.004
      Issue No: Vol. 347 (2017)
  • Nanopteron solutions of diatomic Fermi-Pasta–Ulam-Tsingou lattices
           with small mass-ratio
    • Authors: Aaron Hoffman; J. Douglas Wright
      Abstract: Publication date: Available online 31 July 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Aaron Hoffman, J. Douglas Wright
      Consider an infinite chain of masses, each connected to its nearest neighbors by a (nonlinear) spring. This is a Fermi-Pasta–Ulam-Tsingou lattice. We prove the existence of traveling waves in the setting where the masses alternate in size. In particular we address the limit where the mass ratio tends to zero. The problem is inherently singular and we find that the traveling waves are not true solitary waves but rather “nanopterons”, which is to say, waves which asymptotic at spatial infinity to very small amplitude periodic waves. Moreover, we can only find solutions when the mass ratio lies in a certain open set. The difficulties in the problem all revolve around understanding Jost solutions of a nonlocal Schrödinger operator in its semi-classical limit.

      PubDate: 2017-08-03T21:43:26Z
      DOI: 10.1016/j.physd.2017.07.004
  • Emergence of unstable modes for classical shock waves in isothermal ideal
    • Authors: Heinrich Freistühler; Felix Kleber; Johannes Schropp
      Abstract: Publication date: Available online 31 July 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Heinrich Freistühler, Felix Kleber, Johannes Schropp
      This note studies classical magnetohydrodynamic shock waves in an inviscid fluidic plasma that is assumed to be a perfect conductor of heat as well as of electricity. For this mathematically prototypical material, it identifies, mainly numerically, two critical manifolds in parameter space, across which slow resp. fast MHD shock waves undergo emergence of a complex conjugate pair of unstable transverse modes. For slow shocks, this emergence occurs in a particularly interesting way already in the parallel case, in which it happens at the spectral value λ ˆ ≡ λ ∕ ω = 0 and the critical manifold possesses a simple explicit algebraic representation. Results of refined numerical treatment show that within the set of non-parallel slow shocks the unstable mode pair emerges from two generically different spectral values λ ˆ = ± i γ . For fast shocks, the critical manifold does not intersect the parallel regime and the emergence within the set of non-parallel fast shocks again starts from two generically different spectral values.

      PubDate: 2017-08-03T21:43:26Z
      DOI: 10.1016/j.physd.2017.07.005
  • Time-dependent spectral renormalization method
    • Authors: Justin T. Cole; Ziad H. Musslimani
      Abstract: Publication date: Available online 29 July 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Justin T. Cole, Ziad H. Musslimani
      The spectral renormalization method was introduced by Ablowitz and Musslimani (2005), as an effective way to numerically compute (time-independent) bound states for certain nonlinear boundary value problems. In this paper, we extend those ideas to the time domain and introduce a time-dependent spectral renormalization method as a numerical means to simulate linear and nonlinear evolution equations. The essence of the method is to convert the underlying evolution equation from its partial or ordinary differential form (using Duhamel’s principle) into an integral equation. The solution sought is then viewed as a fixed point in both space and time. The resulting integral equation is then numerically solved using a simple renormalized fixed-point iteration method. Convergence is achieved by introducing a time-dependent renormalization factor which is numerically computed from the physical properties of the governing evolution equation. The proposed method has the ability to incorporate physics into the simulations in the form of conservation laws or dissipation rates. This novel scheme is implemented on benchmark evolution equations: the classical nonlinear Schrödinger (NLS), integrable P T symmetric nonlocal NLS and the viscous Burgers’ equations, each of which being a prototypical example of a conservative and dissipative dynamical system. Numerical implementation and algorithm performance are also discussed.

      PubDate: 2017-08-03T21:43:26Z
      DOI: 10.1016/j.physd.2017.07.006
  • Boundary layer analysis for the stochastic nonlinear
           reaction–diffusion equations
    • Authors: Youngjoon Hong; Chang-Yeol Jung; Roger Temam
      Abstract: Publication date: Available online 29 July 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Youngjoon Hong, Chang-Yeol Jung, Roger Temam
      Singularly perturbed stochastic (and deterministic) nonlinear reaction–diffusion equations are considered. We first study the governing problem posed in the channel domain with lateral periodicity and extend the results to general smooth domains. Introducing corrector functions, which correct the boundary values discrepancies, we are able to develop the convergence analysis. For the analysis, we make use of the maximum principle to estimate the corrector functions. The stochastic problems also rely on the deterministic corrector functions, which lead to simpler computations than those of the stochastic version of the correctors.

      PubDate: 2017-08-03T21:43:26Z
      DOI: 10.1016/j.physd.2017.07.002
  • Multivariate Hadamard self-similarity: Testing fractal connectivity
    • Authors: Herwig Wendt; Gustavo Didier; Sébastien Combrexelle; Patrice Abry
      Abstract: Publication date: Available online 16 July 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Herwig Wendt, Gustavo Didier, Sébastien Combrexelle, Patrice Abry
      While scale invariance is commonly observed in each component of real world multivariate signals, it is also often the case that the inter-component correlation structure is not fractally connected, i.e., its scaling behavior is not determined by that of the individual components. To model this situation in a versatile manner, we introduce a class of multivariate Gaussian stochastic processes called Hadamard fractional Brownian motion (HfBm). Its theoretical study sheds light on the issues raised by the joint requirement of entry-wise scaling and departures from fractal connectivity. An asymptotically normal wavelet-based estimator for its scaling parameter, called the Hurst matrix, is proposed, as well as asymptotically valid confidence intervals. The latter are accompanied by original finite sample procedures for computing confidence intervals and testing fractal connectivity from one single and finite size observation. Monte Carlo simulation studies are used to assess the estimation performance as a function of the (finite) sample size, and to quantify the impact of omitting wavelet cross-correlation terms. The simulation studies are shown to validate the use of approximate confidence intervals, together with the significance level and power of the fractal connectivity test. The test performance and properties are further studied as functions of the HfBm parameters.

      PubDate: 2017-07-24T03:26:34Z
      DOI: 10.1016/j.physd.2017.07.001
  • Integrable U(1)-invariant peakon equations from the NLS hierarchy
    • Authors: Stephen C. Anco; Fatane Mobasheramini
      Abstract: Publication date: Available online 13 July 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Stephen C. Anco, Fatane Mobasheramini
      Two integrable U ( 1 ) -invariant peakon equations are derived from the NLS hierarchy through the tri-Hamiltonian splitting method. A Lax pair, a recursion operator, a bi-Hamiltonian formulation, and a hierarchy of symmetries and conservation laws are obtained for both peakon equations. These equations are also shown to arise as potential flows in the NLS hierarchy by applying the NLS recursion operator to flows generated by space translations and U ( 1 ) -phase rotations on a potential variable. Solutions for both equations are derived using a peakon ansatz combined with an oscillatory temporal phase. This yields the first known example of a peakon breather. Spatially periodic counterparts of these solutions are also obtained.

      PubDate: 2017-07-24T03:26:34Z
      DOI: 10.1016/j.physd.2017.06.006
  • Generic torus canards
    • Authors: Theodore
      Abstract: Publication date: Available online 4 July 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Theodore Vo
      Torus canards are special solutions of fast/slow systems that alternate between attracting and repelling manifolds of limit cycles of the fast subsystem. A relatively new dynamic phenomenon, torus canards have been found in neural applications to mediate the transition from tonic spiking to bursting via amplitude-modulated spiking. In R 3 , torus canards are degenerate: they require one-parameter families of 2-fast/1-slow systems in order to be observed and even then, they only occur on exponentially thin parameter intervals. The addition of a second slow variable unfolds the torus canard phenomenon, making it generic and robust. That is, torus canards in fast/slow systems with (at least) two slow variables occur on open parameter sets. So far, generic torus canards have only been studied numerically, and their behaviour has been inferred based on averaging and canard theory. This approach, however, has not been rigorously justified since the averaging method breaks down near a fold of periodics, which is exactly where torus canards originate. In this work, we combine techniques from Floquet theory, averaging theory, and geometric singular perturbation theory to show that the average of a torus canard is a folded singularity canard. In so doing, we devise an analytic scheme for the identification and topological classification of torus canards in fast/slow systems with two fast variables and k slow variables, for any positive integer k . We demonstrate the predictive power of our results in a model for intracellular calcium dynamics, where we explain the mechanisms underlying a novel class of elliptic bursting rhythms, called amplitude-modulated bursting, by constructing the torus canard analogues of mixed-mode oscillations. We also make explicit the connection between our results here with prior studies of torus canards and torus canard explosion in R 3 , and discuss how our methods can be extended to fast/slow systems of arbitrary (finite) dimension.

      PubDate: 2017-07-12T02:37:34Z
  • Dispersion managed solitons in the presence of saturated nonlinearity
    • Authors: Dirk Hundertmark; Young-Ran Lee; Tobias Ried; Vadim Zharnitsky
      Abstract: Publication date: Available online 29 June 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Dirk Hundertmark, Young-Ran Lee, Tobias Ried, Vadim Zharnitsky
      The averaged dispersion managed nonlinear Schrödinger equation with saturated nonlinearity is considered. It is shown that under rather general assumptions on the saturated nonlinearity, the ground state solution corresponding to the dispersion managed soliton can be found for both zero residual dispersion and positive residual dispersion. The same applies to diffraction management solitons, which are a discrete version describing certain waveguide arrays.

      PubDate: 2017-07-03T11:28:05Z
      DOI: 10.1016/j.physd.2017.06.004
  • Spatiotemporal algebraically localized waveforms for a nonlinear
           Schrödinger model with gain and loss
    • Authors: Z.A. Anastassi; G. Fotopoulos; D.J. Frantzeskakis; T.P. Horikis; N.I. Karachalios; P.G. Kevrekidis; I.G. Stratis; K. Vetas
      Abstract: Publication date: Available online 27 June 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Z.A. Anastassi, G. Fotopoulos, D.J. Frantzeskakis, T.P. Horikis, N.I. Karachalios, P.G. Kevrekidis, I.G. Stratis, K. Vetas
      We consider the asymptotic behavior of the solutions of a nonlinear Schrödinger (NLS) model incorporating linear and nonlinear gain/loss. First, we describe analytically the dynamical regimes (depending on the gain/loss strengths), for finite-time collapse, decay, and global existence of solutions in the dynamics. Then, for all the above parametric regimes, we use direct numerical simulations to study the dynamics corresponding to algebraically decaying initial data. We identify crucial differences between the dynamics of vanishing initial conditions, and those converging to a finite constant background: in the former (latter) case we find strong (weak) collapse or decay, when the gain/loss parameters are selected from the relevant regimes. One of our main results, is that in all the above regimes, non-vanishing initial data transition through spatiotemporal, algebraically decaying waveforms. While the system is nonintegrable, the evolution of these waveforms is reminiscent to the evolution of the Peregrine rogue wave of the integrable NLS limit. The parametric range of gain and loss for which this phenomenology persists is also touched upon.

      PubDate: 2017-07-03T11:28:05Z
      DOI: 10.1016/j.physd.2017.06.003
  • Overhanging of membranes and filaments adhering to periodic graph
    • Authors: Tatsuya Miura
      Abstract: Publication date: Available online 20 June 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Tatsuya Miura
      This paper mathematically studies membranes and filaments adhering to periodic patterned substrates in a one-dimensional model. The problem is formulated by the minimizing problem of an elastic energy with a contact potential on graph substrates. Global minimizers (ground states) are mainly considered in view of their graph representations. Our main results exhibit sufficient conditions for the graph representation and examples of situations where any global minimizer must overhang.

      PubDate: 2017-06-22T11:05:22Z
  • Distributed synaptic weights in a LIF neural network and learning rules
    • Authors: Benoî t Perthame; Delphine Salort; Gilles Wainrib
      Abstract: Publication date: Available online 15 June 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Benoî t Perthame, Delphine Salort, Gilles Wainrib
      Leaky integrate-and-fire (LIF) models are mean-field limits, with a large number of neurons, used to describe neural networks. We consider inhomogeneous networks structured by a connectivity parameter (strengths of the synaptic weights) with the effect of processing the input current with different intensities. We first study the properties of the network activity depending on the distribution of synaptic weights and in particular its discrimination capacity. Then, we consider simple learning rules and determine the synaptic weight distribution it generates. We outline the role of noise as a selection principle and the capacity to memorized a learned signal.

      PubDate: 2017-06-16T10:48:06Z
      DOI: 10.1016/j.physd.2017.05.005
  • Multi-model cross-pollination in time
    • Authors: Hailiang Du; Leonard A. Smith
      Abstract: Publication date: Available online 13 June 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Hailiang Du, Leonard A. Smith
      The predictive skill of complex models is rarely uniform in model-state space; in weather forecasting models, for example, the skill of the model can be greater in the regions of most interest to a particular operational agency than it is in “remote” regions of the globe. Given a collection of models, a multi-model forecast system using the cross-pollination in time approach can be generalized to take advantage of instances where some models produce forecasts with more information regarding specific components of the model-state than other models, systematically. This generalization is stated and then successfully demonstrated in a moderate ( ∼ 40 ) dimensional nonlinear dynamical system, suggested by Lorenz, using four imperfect models with similar global forecast skill. Applications to weather forecasting and in economic forecasting are discussed. Given that the relative importance of different phenomena in shaping the weather changes in latitude, changes in attitude among forecast centers in terms of the resources assigned to each phenomena are to be expected. The demonstration establishes that cross-pollinating elements of forecast trajectories enriches the collection of simulations upon which the forecast is built, and given the same collection of models can yield a new forecast system with significantly more skill than the original forecast system.

      PubDate: 2017-06-16T10:48:06Z
      DOI: 10.1016/j.physd.2017.06.001
  • Well-posedness and dynamics of a fractional stochastic
           integro-differential equation
    • Authors: Linfang Liu; Tomás Caraballo
      Abstract: Publication date: Available online 9 June 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Linfang Liu, Tomás Caraballo
      In this paper we investigate the well-posedness and dynamics of a fractional stochastic integro-differential equation describing a reaction process depending on the temperature itself. Existence and uniqueness of solutions of the integro-differential equation is proved by the Lumer–Phillips theorem. Besides, under appropriate assumptions on the memory kernel and on the magnitude of the nonlinearity, the existence of random attractor is achieved by obtaining first some a priori estimates. Moreover, the random attractor is shown to have finite Hausdorff dimension.

      PubDate: 2017-06-12T10:36:04Z
      DOI: 10.1016/j.physd.2017.05.006
  • Analysis of mixed-mode oscillation-incrementing bifurcations generated in
           a nonautonomous constrained Bonhoeffer–van der Pol oscillator
    • Authors: Takuji Kousaka; Yutsuki Ogura; Kuniyasu Shimizu; Hiroyuki Asahara; Naohiko Inaba
      Abstract: Publication date: Available online 1 June 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Takuji Kousaka, Yutsuki Ogura, Kuniyasu Shimizu, Hiroyuki Asahara, Naohiko Inaba
      Mixed-mode oscillations (MMOs) are phenomena observed in a number of dynamic settings, including electrical circuits and chemical systems. Mixed-mode oscillation-incrementing bifurcations (MMOIBs) are among the most complex MMO bifurcations observed in the large group of MMO-generating dynamics; however, only a few theoretical analyses of the mechanism causing MMOIBs have been performed to date. In this study, we use a degenerate technique to analyze MMOIBs generated in a Bonhoeffer-van der Pol oscillator with a diode under weak periodic perturbation. We consider the idealized case in which the diode operates as an ideal switch; in this case, the governing equation of the oscillator is a piecewise smooth constraint equation and the Poincaré return map is one-dimensional, and we find that MMOIBs occur in a manner similar to period-adding bifurcations generated by the circle map. Our numerical results suggest that the universal constant converges to 1.0 and our experimental results demonstrate that MMOIBs can occur successively many times. Our one-dimensional Poincaré return map clearly answers the fundamental question of why MMOs are related to Farey sequences even though each MMO-generating region in the parameter space is terminated by chaos.

      PubDate: 2017-06-02T09:28:10Z
      DOI: 10.1016/j.physd.2017.05.001
  • Averaging theory at any order for computing limit cycles of discontinuous
           piecewise differential systems with many zones
    • Authors: Jaume Llibre; Douglas D. Novaes; Camila A.B. Rodrigues
      Abstract: Publication date: Available online 24 May 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Jaume Llibre, Douglas D. Novaes, Camila A.B. Rodrigues
      This work is devoted to study the existence of periodic solutions for a class ε -family of discontinuous differential systems with many zones. We show that the averaged functions at any order control the existence of crossing limit cycles for systems in this class. We also provide some examples dealing with nonsmooth perturbations of nonlinear centers.

      PubDate: 2017-05-28T09:23:39Z
      DOI: 10.1016/j.physd.2017.05.003
  • The influence of canalization on the robustness of Boolean networks
    • Authors: C. Kadelka; J. Kuipers; R. Laubenbacher
      Abstract: Publication date: Available online 17 May 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): C. Kadelka, J. Kuipers, R. Laubenbacher
      Time- and state-discrete dynamical systems are frequently used to model molecular networks. This paper provides a collection of mathematical and computational tools for the study of robustness in Boolean network models. The focus is on networks governed by k -canalizing functions, a recently introduced class of Boolean functions that contains the well-studied class of nested canalizing functions. The variable activities and sensitivity of a function quantify the impact of input changes on the function output. This paper generalizes the latter concept to c -sensitivity and provides formulas for the activities and c -sensitivity of general k -canalizing functions as well as canalizing functions with more precisely defined structure. A popular measure for the robustness of a network, the Derrida value, can be expressed as a weighted sum of the c -sensitivities of the governing canalizing functions, and can also be calculated for a stochastic extension of Boolean networks. These findings provide a computationally efficient way to obtain Derrida values of Boolean networks, deterministic or stochastic, that does not involve simulation.

      PubDate: 2017-05-23T09:10:50Z
      DOI: 10.1016/j.physd.2017.05.002
  • Optimal strategies for the control of autonomous vehicles in data
    • Authors: D. McDougall; R.O. Moore
      Abstract: Publication date: Available online 5 May 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): D. McDougall, R.O. Moore
      We propose a method to compute optimal control paths for autonomous vehicles deployed for the purpose of inferring a velocity field. In addition to being advected by the flow, the vehicles are able to effect a fixed relative speed with arbitrary control over direction. It is this direction that is used as the basis for the locally optimal control algorithm presented here, with objective formed from the variance trace of the expected posterior distribution. We present results for linear flows near hyperbolic fixed points.

      PubDate: 2017-05-07T15:23:30Z
      DOI: 10.1016/j.physd.2017.04.001
  • Solution landscapes in nematic microfluidics
    • Authors: M. Crespo; A. Majumdar; A.M. Ramos; I.M. Griffiths
      Abstract: Publication date: Available online 4 May 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): M. Crespo, A. Majumdar, A.M. Ramos, I.M. Griffiths
      We study the static equilibria of a simplified Leslie–Ericksen model for a unidirectional uniaxial nematic flow in a prototype microfluidic channel, as a function of the pressure gradient G and inverse anchoring strength, B . We numerically find multiple static equilibria for admissible pairs ( G , B ) and classify them according to their winding numbers and stability. The case G = 0 is analytically tractable and we numerically study how the solution landscape is transformed as G increases. We study the one-dimensional dynamical model, the sensitivity of the dynamic solutions to initial conditions and the rate of change of G and B . We provide a physically interesting example of how the time delay between the applications of G and B can determine the selection of the final steady state.

      PubDate: 2017-05-07T15:23:30Z
      DOI: 10.1016/j.physd.2017.04.004
  • Invariant manifolds and the parameterization method in coupled energy
           harvesting piezoelectric oscillators
    • Authors: Albert Granados
      Abstract: Publication date: Available online 19 April 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Albert Granados
      Energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy. Widely extended are those based on piezoelectric materials, whose dynamics are Hamiltonian submitted to different sources of dissipation: damping and coupling. These dissipations bring the system to low energy regimes, which is not desired in long term as it diminishes the absorbed energy. To avoid or to minimize such situations, we propose that the coupling of two oscillators could benefit from theory of Arnold diffusion. Such phenomenon studies O ( 1 ) energy variations in Hamiltonian systems and hence could be very useful in energy harvesting applications. This article is a first step towards this goal. We consider two piezoelectric beams submitted to a small forcing and coupled through an electric circuit. By considering the coupling, damping and forcing as perturbations, we prove that the unperturbed system possesses a 4-dimensional Normally Hyperbolic Invariant Manifold with 5 and 4-dimensional stable and unstable manifolds, respectively. These are locally unique after the perturbation. By means of the parameterization method, we numerically compute parameterizations of the perturbed manifold, its stable and unstable manifolds and study its inner dynamics. We show evidence of homoclinic connections when the perturbation is switched on.

      PubDate: 2017-04-25T10:34:15Z
      DOI: 10.1016/j.physd.2017.04.003
  • The spectrum of the torus profile to a geometric variational problem with
           long range interaction
    • Authors: Xiaofeng Ren; Juncheng Wei
      Abstract: Publication date: Available online 18 April 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Xiaofeng Ren, Juncheng Wei
      The profile problem for the Ohta-Kawasaki diblock copolymer theory is a geometric variational problem. The energy functional is defined on sets in R 3 of prescribed volume and the energy of an admissible set is its perimeter plus a long range interaction term related to the Newtonian potential of the set. This problem admits a solution, called a torus profile, that is a set enclosed by an approximate torus of the major radius 1 and the minor radius q . The torus profile is both axially symmetric about the z axis and reflexively symmetric about the x y -plane. There is a way to set up the profile problem in a function space as a partial differential-integro equation. The linearized operator L of the problem at the torus profile is decomposed into a family of linear ordinary differential-integro operators L m where the index m = 0 , 1 , 2 , . . . is called a mode. The spectrum of L is the union of the spectra of the L m ’s. It is proved that for each m , when q is sufficiently small, L m is positive definite. ( 0 is an eigenvalue for both L 0 and L 1 , due to the translation and rotation invariance.) As q tends to 0 , more and more L m ’s become positive definite. However no matter how small q is, there is always a mode m of which L m has a negative eigenvalue. This mode grows to infinity like q − 3 / 4 as q → 0 .

      PubDate: 2017-04-25T10:34:15Z
      DOI: 10.1016/j.physd.2017.01.001
  • Dynamical and energetic instabilities of F=2 spinor Bose-Einstein
           condensates in an optical lattice
    • Authors: Deng-Shan Wang; Yu-Ren Shi; Wen-Xing Feng; Lin Wen
      Abstract: Publication date: Available online 17 April 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Deng-Shan Wang, Yu-Ren Shi, Wen-Xing Feng, Lin Wen
      The dynamical and energetic instabilities of the F =2 spinor Bose-Einstein condensates in an optical lattice are investigated theoretically and numerically. By analyzing the dynamical response of different carrier waves to an additional linear perturbation, we obtain the instability criteria for the ferromagnetic, uniaxial nematic, biaxial nematic and cyclic states, respectively. When an external magnetic field is taken into account, we find that the linear or quadratic Zeeman effects obviously affect the dynamical instability properties of uniaxial nematic, biaxial nematic and cyclic states, but not for the ferromagnetic one. In particular, it is found that the faster moving F =2 spinor BEC has a larger energetic instability region than lower one in all the four states. In addition, it is seen that for most states there probably exists a critical value k c > 0 , for which k > k c gives the energetic instability to arise under appreciative parameters.

      PubDate: 2017-04-18T03:43:03Z
      DOI: 10.1016/j.physd.2017.04.002
  • On the phenomenon of mixed dynamics in Pikovsky-Topaj system of coupled
    • Authors: A.S. Gonchenko; S.V. Gonchenko; A.O. Kazakov; D.V. Turaev
      Abstract: Publication date: Available online 30 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): A.S. Gonchenko, S.V. Gonchenko, A.O. Kazakov, D.V. Turaev
      A one-parameter family of time-reversible systems on three-dimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits and the closure of the completely unstable periodic orbits. We search for the stable and unstable periodic orbits indirectly, by finding non-conservative saddle periodic orbits and heteroclinic connections between them. In this way, we are able to claim the existence of mixed dynamics for a large range of parameter values. We investigate local and global bifurcations that can be used for the detection of mixed dynamics.

      PubDate: 2017-04-04T03:13:49Z
      DOI: 10.1016/j.physd.2017.02.002
  • Pattern formation on the free surface of a ferrofluid: Spatial dynamics
           and homoclinic bifurcation
    • Authors: M.D. Groves; D.J.B. Lloyd; A. Stylianou
      Abstract: Publication date: Available online 23 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): M.D. Groves, D.J.B. Lloyd, A. Stylianou
      We establish the existence of spatially localised one-dimensional free surfaces of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetisation law. It is shown that the ferrohydrostatic equations can be derived from a variational principle that allows one to formulate them as an (infinite-dimensional) spatial Hamiltonian system in which the unbounded free-surface direction plays the role of time. A centre-manifold reduction technique converts the problem for small solutions near onset to an equivalent Hamiltonian system with finitely many degrees of freedom. Normal-form theory yields the existence of homoclinic solutions to the reduced system, which correspond to spatially localised solutions of the ferrohydrostatic equations.

      PubDate: 2017-03-28T02:42:26Z
      DOI: 10.1016/j.physd.2017.03.004
  • Synchronization of weakly coupled canard oscillators
    • Authors: Elif Köksal Ersöz; Mathieu Desroches; Martin Krupa
      Abstract: Publication date: Available online 22 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Elif Köksal Ersöz, Mathieu Desroches, Martin Krupa
      Synchronization has been studied extensively in the context of weakly coupled oscillators using the so-called phase response curve (PRC) which measures how a change of the phase of an oscillator is affected by a small perturbation. This approach was based upon the work of Malkin, and it has been extended to relaxation oscillators. Namely, synchronization conditions were established under the weak coupling assumption, leading to a criterion for the existence of synchronous solutions of weakly coupled relaxation oscillators. Previous analysis relies on the fact that the slow nullcline does not intersect the fast nullcline near one of its fold points, where canard solutions can arise. In the present study we use numerical continuation techniques to solve the adjoint equations and we show that synchronization properties of canard cycles are different than those of classical relaxation cycles. In particular, we highlight a new special role of the maximal canard in separating two distinct synchronization regimes: the Hopf regime and the relaxation regime. Phase plane analysis of slow-fast oscillators undergoing a canard explosion provides an explanation for this change of synchronization properties across the maximal canard.

      PubDate: 2017-03-28T02:42:26Z
      DOI: 10.1016/j.physd.2017.02.016
  • Integrable systems and invariant curve flows in symplectic Grassmannian
    • Authors: Junfeng Song; Changzheng Qu; Ruoxia Yao
      Abstract: Publication date: Available online 21 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Junfeng Song, Changzheng Qu, Ruoxia Yao
      In this paper, local geometry of curves in the symplectic Grassmannian homogeneous space Sp ( 4 , R ) / ( Sp ( 2 , R ) × Sp ( 2 , R ) ) and its connection with that of the pseudo-hyperbolic space H 2 , 2 are studied. The group-based Serret-Frenet equations and the associated Maurer-Cartan differential invariants for the Grassmannian curves are obtained by using the equivariant moving frame method. The Grassmannian natural frame are also constructed by a gauge transformation from the Serret-Frenet frame, relating to the hyperbolic natural frame by the local Lie group isomorphism. Using the natural frames, invariant curve flows in the Grassmannian and the hyperbolic spaces are studied. It is shown that certain intrinsic curve flows induce the bi-Hamiltonian integrable matrix mKdV equation on the Maurer-Cartan differential invariants.

      PubDate: 2017-03-28T02:42:26Z
      DOI: 10.1016/j.physd.2017.02.013
  • Stability of the phase motion in race-track microtons
    • Authors: Yu.A. Kubyshin; O. Larreal; R. Ramírez-Ros; T.M. Seara
      Abstract: Publication date: Available online 18 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Yu.A. Kubyshin, O. Larreal, R. Ramírez-Ros, T.M. Seara
      We model the phase oscillations of electrons in race-track microtrons by means of an area preserving map with a fixed point at the origin, which represents the synchronous trajectory of a reference particle in the beam. We study the nonlinear stability of the origin in terms of the synchronous phase —the phase of the synchronous particle at the injection. We estimate the size and shape of the stability domain around the origin, whose main connected component is enclosed by an invariant curve. We describe the evolution of the stability domain as the synchronous phase varies. We also clarify the role of the stable and unstable invariant curves of some hyperbolic (fixed or periodic) points.

      PubDate: 2017-03-20T19:31:07Z
      DOI: 10.1016/j.physd.2017.03.001
  • Markovian properties of velocity increments in boundary layer turbulence
    • Authors: Murat Tutkun
      Abstract: Publication date: Available online 16 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Murat Tutkun
      Markovian properties of the turbulent velocity increments in a flat plate boundary layer at Re θ = 19100 are investigated using hot-wire anemometry measurements of the streamwise velocity component in a wind tunnel. Increments of the longitudinal velocities at different wall-normal positions show that the flow exhibits Markovian properties when the separation between different scales, or the Markov-Einstein coherence length, is on the order of the Taylor microscale, λ . The results indicate that Markovian nature of turbulence evolves across the boundary layer showing certain characteristics pertaining to the distance to the wall. The connection between the Markovian properties of turbulent boundary layer and existence of the spectral gap is explored. Markovianity of the process is also discussed in relation to the nonlocal nonlinear versus local nonlinear transfer of energy, triadic interactions and dissipation.

      PubDate: 2017-03-20T19:31:07Z
      DOI: 10.1016/j.physd.2017.03.002
  • Targeted energy transfer in laminar vortex-induced vibration of a sprung
           cylinder with a nonlinear dissipative rotator
    • Authors: Antoine Blanchard; Lawrence A. Bergman; Alexander F. Vakakis
      Abstract: Publication date: Available online 16 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Antoine Blanchard, Lawrence A. Bergman, Alexander F. Vakakis
      We computationally investigate the dynamics of a linearly-sprung circular cylinder immersed in an incompressible flow and undergoing transverse vortex-induced vibration (VIV), to which is attached a rotational nonlinear energy sink (NES) consisting of a mass that freely rotates at constant radius about the cylinder axis, and whose motion is restrained by a rotational linear viscous damper. The inertial coupling between the rotational motion of the attached mass and the rectilinear motion of the cylinder is “essentially nonlinear”, which, in conjunction with dissipation, allows for one-way, nearly irreversible targeted energy transfer (TET) from the oscillating cylinder to the nonlinear dissipative attachment. At the intermediate Reynolds number R e = 100 , the NES-equipped sprung cylinder undergoes repetitive cycles of slowly decaying oscillations punctuated by intervals of chaotic instabilities. During the slowly decaying portion of each cycle, the dynamics of the cylinder is regular and, for large enough values of the ratio ε of the NES mass to the total mass (i.e., NES mass plus cylinder mass), can lead to significant vortex street elongation with partial stabilization of the wake. As ε approaches zero, no such vortex elongation is observed and the wake patterns appear similar to that for a sprung cylinder with no NES. We apply proper orthogonal decomposition (POD) to the velocity flow field during a slowly decaying portion of the solution and show that, in situations where vortex elongation occurs, the NES, though not in direct contact with the surrounding fluid, has a drastic effect on the underlying flow structures, imparting significant and continuous passive redistribution of energy among POD modes. We construct a POD-based reduced-order model for the lift coefficient to characterize energy transactions between the fluid and the cylinder throughout the slowly decaying cycle. We introduce a quantitative signed measure of the work done by the fluid on the cylinder over one quasi-period of the slowly decaying response and find that vortex elongation is associated with a sign change of that measure, indicating that a reversal of the direction of energy transfer, with the cylinder “leaking energy back” to the flow, is responsible for partial stabilization and elongation of the wake. We interpret these findings in terms of the spatial structure and energy distribution of the POD modes, and relate them to the mechanism of transient resonance capture into a slow invariant manifold of the fluid–structure interaction dynamics.

      PubDate: 2017-03-20T19:31:07Z
      DOI: 10.1016/j.physd.2017.03.003
  • Fluctuations, response, and resonances in a simple atmospheric model
    • Authors: Andrey Gritsun; Valerio Lucarini
      Abstract: Publication date: Available online 14 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Andrey Gritsun, Valerio Lucarini
      We study the response of a simple quasi-geostrophic barotropic model of the atmosphere to various classes of perturbations affecting its forcing and its dissipation using the formalism of the Ruelle response theory. We investigate the geometry of such perturbations by constructing the covariant Lyapunov vectors of the unperturbed system and discover in one specific case–orographic forcing–a substantial projection of the forcing onto the stable directions of the flow. This results into a resonant response shaped as a Rossby-like wave that has no resemblance to the unforced variability in the same range of spatial and temporal scales. Such a climatic surprise corresponds to a violation of the fluctuation-dissipation theorem, in agreement with the basic tenets of nonequilibrium statistical mechanics. The resonance can be attributed to a specific group of rarely visited unstable periodic orbits of the unperturbed system. Our results reinforce the idea of using basic methods of nonequilibrium statistical mechanics and high-dimensional chaotic dynamical systems to approach the problem of understanding climate dynamics.

      PubDate: 2017-03-20T19:31:07Z
      DOI: 10.1016/j.physd.2017.02.015
  • Standing and travelling waves in a spherical brain model: The Nunez model
    • Authors: S. Visser; R. Nicks; O. Faugeras; S. Coombes
      Abstract: Publication date: Available online 8 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): S. Visser, R. Nicks, O. Faugeras, S. Coombes
      The Nunez model for the generation of electroencephalogram (EEG) signals is naturally described as a neural field model on a sphere with space-dependent delays. For simplicity, dynamical realisations of this model either as a damped wave equation or an integro-differential equation, have typically been studied in idealised one dimensional or planar settings. Here we revisit the original Nunez model to specifically address the role of spherical topology on spatio-temporal pattern generation. We do this using a mixture of Turing instability analysis, symmetric bifurcation theory, center manifold reduction and direct simulations with a bespoke numerical scheme. In particular we examine standing and travelling wave solutions using normal form computation of primary and secondary bifurcations from a steady state. Interestingly, we observe spatio-temporal patterns which have counterparts seen in the EEG patterns of both epileptic and schizophrenic brain conditions.

      PubDate: 2017-03-12T21:31:41Z
      DOI: 10.1016/j.physd.2017.02.017
  • Controlling coexisting attractors of an impacting system via linear
    • Authors: Yang Liu; Joseph Páez Chávez
      Abstract: Publication date: Available online 8 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Yang Liu, Joseph Páez Chávez
      This paper studies the control of coexisting attractors in an impacting system via a recently developed control law based on linear augmentation. Special attention is given to two control issues in the framework of multistable engineering systems, namely, the switching between coexisting attractors without altering the system’s main parameters and the avoidance of grazing-induced chaotic responses. The effectiveness of the proposed control scheme is confirmed numerically for the case of a periodically excited, soft impact oscillator. Our analysis shows how path-following techniques for non-smooth systems can be used in order to determine the optimal control parameters in terms of energy expenditure due to the control signal and transient behavior of the control error, which can be applied to a broad range of engineering problems.

      PubDate: 2017-03-12T21:31:41Z
      DOI: 10.1016/j.physd.2017.02.018
  • Construction of Darboux coordinates and Poincaré-Birkhoff normal forms in
           noncanonical Hamiltonian systems
    • Authors: Andrej Junginger; Jörg Main; Günter Wunner
      Abstract: Publication date: Available online 7 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Andrej Junginger, Jörg Main, Günter Wunner
      We demonstrate a general method to construct Darboux coordinates via normal form expansions in noncanonical Hamiltonian system obtained from e.g. a variational approach to quantum systems. The procedure serves as a tool to naturally extract canonical coordinates out of the variational parameters and at the same time to transform the energy functional into its Poincaré-Birkhoff normal form. The method is general in the sense that it is applicable for arbitrary degrees of freedom, in arbitrary orders of the local expansion, and it is independent of the precise form of the Hamilton operator. The method presented allows for the general and systematic investigation of quantum systems in the vicinity of fixed points, which e.g. correspond to ground, excited or transition states. Moreover, it directly allows to calculate classical and quantum reaction rates by applying transition state theory.

      PubDate: 2017-03-12T21:31:41Z
      DOI: 10.1016/j.physd.2017.02.014
  • Assessment of the effects of azimuthal mode number perturbations upon the
           implosion processes of fluids in cylinders
    • Authors: Michael Lindstrom
      Abstract: Publication date: Available online 1 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Michael Lindstrom
      Fluid instabilities arise in a variety of contexts and are often unwanted results of engineering imperfections. In one particular model for a magnetized target fusion reactor, a pressure wave is propagated in a cylindrical annulus comprised of a dense fluid before impinging upon a plasma and imploding it. Part of the success of the apparatus is a function of how axially-symmetric the final pressure pulse is upon impacting the plasma. We study a simple model for the implosion of the system to study how imperfections in the pressure imparted on the outer circumference grow due to geometric focusing. Our methodology entails linearizing the compressible Euler equations for mass and momentum conservation about a cylindrically symmetric problem and analyzing the perturbed profiles at different mode numbers. The linearized system gives rise to singular shocks and through analyzing the perturbation profiles at various times, we infer that high mode numbers are dampened through the propagation. We also study the Linear Klein-Gordon equation in the context of stability of linear cylindrical wave formation whereby highly oscillatory, bounded behaviour is observed in a far field solution.

      PubDate: 2017-03-07T21:14:06Z
      DOI: 10.1016/j.physd.2017.02.012
  • KdV cnoidal waves in a traffic flow model with periodic boundaries
    • Authors: L.L. Hattam
      Abstract: Publication date: Available online 1 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): L.L. Hattam
      An optimal-velocity (OV) model describes car motion on a single lane road. In particular, near to the boundary signifying the onset of traffic jams, this model reduces to a perturbed Korteweg-de Vries (KdV) equation using asymptotic analysis. Previously, the KdV soliton solution has then been found and compared to numerical results (see Muramatsu and Nagatani (1999)). Here, we instead apply modulation theory to this perturbed KdV equation to obtain at leading order, the modulated cnoidal wave solution. At the next order, the Whitham equations are derived, which have been modified due to the equation perturbation terms. Next, from this modulation system, a family of spatially periodic cnoidal waves are identified that characterise vehicle headway distance. Then, for this set of solutions, we establish the relationship between the wave speed, the modulation term and the driver sensitivity. This analysis is confirmed with comparisons to numerical solutions of the OV model. As well, the long-time behaviour of these solutions is investigated.

      PubDate: 2017-03-07T21:14:06Z
      DOI: 10.1016/j.physd.2017.02.010
  • Controlling roughening processes in the stochastic Kuramoto-Sivashinsky
    • Authors: S.N. Gomes; S. Kalliadasis; D.T. Papageorgiou; G.A. Pavliotis; M. Pradas
      Abstract: Publication date: Available online 1 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): S.N. Gomes, S. Kalliadasis, D.T. Papageorgiou, G.A. Pavliotis, M. Pradas
      We present a novel control methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strategy is then based on two steps: first, stabilize the zero solution of the deterministic part and, second, control the roughness of the stochastic linear equation. We consider both periodic controls and point actuated ones, observing in all cases that the second moment of the solution evolves in time according to a power-law until it saturates at the desired controlled value.

      PubDate: 2017-03-07T21:14:06Z
      DOI: 10.1016/j.physd.2017.02.011
  • A novel route to a Hopf bifurcation scenario in switched systems with
    • Authors: P. Kowalczyk
      Abstract: Publication date: Available online 1 March 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): P. Kowalczyk
      Planar switched systems with dead-zone are analyzed. In particular, we consider the effects of a perturbation which is applied to a linear control law and, due to the perturbation, the control changes from purely positional to position-velocity control. This type of a perturbation leads to a novel Hopf-like discontinuity induced bifurcation. We show that this bifurcation leads to the creation of a small scale limit cycle attractor, which scales as the square root of the bifurcation parameter. We then investigate numerically a planar switched system with a positional feedback law, dead-zone and time delay in the switching function. Using the same parameter values as for the switched system without time delay in the switching function, we show a Hopf-like bifurcation scenario which exhibits a qualitative and a quantitative agreement with the scenario analyzed for the non-delayed system.

      PubDate: 2017-03-07T21:14:06Z
      DOI: 10.1016/j.physd.2017.02.007
  • Hopf and Homoclinic Bifurcations on the sliding vector field of switching
           systems in R3: A case study in power electronics
    • Authors: Rony Cristiano; Tiago Carvalho; Durval J. Tonon; Daniel J. Pagano
      Abstract: Publication date: Available online 24 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Rony Cristiano, Tiago Carvalho, Durval J. Tonon, Daniel J. Pagano
      In this paper, Hopf and homoclinic bifurcations that occur in the sliding vector field of switching systems in R 3 are studied. In particular, a dc-dc boost converter with sliding mode control and washout filter is analyzed. This device is modelled as a three-dimensional Filippov system, characterized by the existence of sliding movement and restricted to the switching manifold. The operating point of the converter is a stable pseudo-equilibrium and it undergoes a subcritical Hopf bifurcation. Such a bifurcation occurs in the sliding vector field and creates, in this field, an unstable limit cycle. The limit cycle is connected to the switching manifold and disappears when it touches the visible-invisible two-fold point, resulting in an homoclinic loop which itself closes in this two-fold point. The study of these dynamic phenomena that can be found in different power electronic circuits controlled by sliding mode control strategies are relevant from the viewpoint of the global stability and robustness of the control design.

      PubDate: 2017-03-01T05:09:55Z
      DOI: 10.1016/j.physd.2017.02.005
  • Axisymmetric pulse train solutions in narrow-gap spherical Couette flow
    • Authors: Adam Child; Rainer Hollerbach; Evy Kersalé
      Abstract: Publication date: Available online 24 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Adam Child, Rainer Hollerbach, Evy Kersalé
      We numerically compute the flow induced in a spherical shell by fixing the outer sphere and rotating the inner one. The aspect ratio ϵ = ( r o − r i ) / r i is set at 0.04 and 0.02, and in each case the Reynolds number measuring the inner sphere’s rotation rate is increased to ∼ 10% beyond the first bifurcation from the basic state flow. For ϵ = 0.04 the initial bifurcations are the same as in previous numerical work at ϵ = 0.154 , and result in steady one- and two-vortex states. Further bifurcations yield travelling wave solutions similar to previous analytic results valid in the ϵ → 0 limit. For ϵ = 0.02 the steady one-vortex state no longer exists, and the first bifurcation is directly to these travelling wave solutions, consisting of pulse trains of Taylor vortices travelling toward the equator from both hemispheres, and annihilating there in distinct phase-slip events. We explore these time-dependent solutions in detail, and find that they can be both equatorially symmetric and asymmetric, as well as periodic or quasi-periodic in time.

      PubDate: 2017-03-01T05:09:55Z
      DOI: 10.1016/j.physd.2017.02.009
  • Complex predator invasion waves in a Holling-Tanner model with nonlocal
           prey interaction
    • Authors: A. Bayliss; V.A. Volpert
      Abstract: Publication date: Available online 20 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): A. Bayliss, V.A. Volpert
      We consider predator invasions for the nonlocal Holling-Tanner model. Predators are introduced in a small region adjacent to an extensive predator-free region. In its simplest form an invasion front propagates into the predator-free region with a predator-prey coexistence state displacing the predator-free state. However, patterns may form in the wake of the invasion front due to instability of the coexistence state. The coexistence state can be subject to either oscillatory or cellular instability, depending on parameters. Furthermore, the oscillatory instability can be either at zero wave number or finite wave number. In addition, the (unstable) predator-free state can be subject to additional cellular instabilities when the extent of the nonlocality is sufficiently large. We perform numerical simulations that demonstrate that the invasion wave may have a complex structure in which different spatial regions exhibit qualitatively different behaviors. These regions are separated by relatively narrow transition regions that we refer to as fronts. We also derive analytic approximations for the speeds of the fronts and find qualitative and quantitative agreement with the results of computations.

      PubDate: 2017-02-21T09:49:54Z
      DOI: 10.1016/j.physd.2017.02.003
  • Elementary solutions for a model Boltzmann equation in one dimension and
           the connection to grossly determined solutions
    • Authors: Thomas E. Carty
      Abstract: Publication date: Available online 20 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Thomas E. Carty
      The Fourier-transformed version of the time dependent slip-flow model Boltzmann equation associated with the linearized BGK model is solved in order to determine the solution’s asymptotics. The ultimate goal of this paper is to demonstrate that there exists a robust set of solutions to this model Boltzmann equation that possess a special property that was conjectured by Truesdell and Muncaster: that solutions decay to a subclass of the solution set uniquely determined by the initial mass density of the gas called the grossly determined solutions. First we determine the spectrum and eigendistributions of the associated homogeneous equation. Then, using Case’s method of elementary solutions, we find analytic time-dependent solutions to the model Boltzmann equation for initial data with a specialized compact support condition under the Fourier transform. In doing so, we show that the spectrum separates the solutions into two distinct parts: one that behaves as a set of transient solutions and the other limiting to a stable subclass of solutions. Thus, we demonstrate that for gas flows with this specialized initial density condition, in time all gas flows for the one dimensional model Boltzmann equation act as grossly determined solutions.

      PubDate: 2017-02-21T09:49:54Z
      DOI: 10.1016/j.physd.2017.02.008
  • Bright and dark solitons in the unidirectional long wave limit for the
           energy transfer on anharmonic crystal lattices
    • Authors: Luis A. Cisneros-Ake; José F. Solano Peláez
      Abstract: Publication date: Available online 9 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Luis A. Cisneros-Ake, José F. Solano Peláez
      The problem of energy transportation along a cubic anharmonic crystal lattice, in the unidirectional long wave limit, is considered. A detailed process, in the discrete lattice equations, shows that unidirectional stable propagating waves for the continuum limit produces a coupled system between a nonlinear Schrödinger (NLS) equation and the Korteweg-deVries (KdV) equation. The traveling wave formalism provides a diversity of exact solutions ranging from the classical Davydov’s soliton (subsonic and supersonic) of the first and second kind to a class consisting in the coupling between the KdV soliton and dark solitons containing the typical ones (similar to the dark-gray soliton in the standard defocusing NLS) and a new kind in the form of a two-hump dark soliton. This family of exact solutions are numerically tested, by means of the pseudo spectral method, in our NLS-KdV system.

      PubDate: 2017-02-15T09:34:42Z
      DOI: 10.1016/j.physd.2017.02.001
  • The Lyapunov-Krasovskii theorem and a sufficient criterion for local
           stability of isochronal synchronization in networks of delay-coupled
    • Authors: J.M.V. Grzybowski; E.E.N. Macau; T. Yoneyama
      Abstract: Publication date: Available online 9 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): J.M.V. Grzybowski, E.E.N. Macau, T. Yoneyama
      This paper presents a self-contained framework for the stability assessment of isochronal synchronization in networks of chaotic and limit-cycle oscillators. The results were based on the Lyapunov-Krasovskii theorem and they establish a sufficient condition for local synchronization stability of as a function of the system and network parameters. With this in mind, a network of mutually delay-coupled oscillators subject to direct self-coupling is considered and then the resulting error equations are block-diagonalized for the purpose of studying their stability. These error equations are evaluated by means of analytical stability results derived from the Lyapunov-Krasovskii theorem. The proposed approach is shown to be a feasible option for the investigation of local stability of isochronal synchronization for a variety of oscillators coupled through linear functions of the state variables under a given undirected graph structure. This ultimately permits the systematic identification of stability regions within the high-dimensionality of the network parameter space. Examples of applications of the results to a number of networks of delay-coupled chaotic and limit-cycle oscillators are provided, such as Lorenz, Rössler, Cubic Chua’s circuit, Van der Pol oscillator and the Hindmarsh-Rose neuron.

      PubDate: 2017-02-15T09:34:42Z
      DOI: 10.1016/j.physd.2017.01.005
  • Wave fronts and cascades of soliton interactions in the periodic two
           dimensional Volterra system
    • Authors: Rhys Bury; Alexander V. Mikhailov; Jing Ping Wang
      Abstract: Publication date: Available online 6 February 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Rhys Bury, Alexander V. Mikhailov, Jing Ping Wang
      In the paper we develop the dressing method for the solution of the two-dimensional periodic Volterra system with a period N . We derive soliton solutions of arbitrary rank k and give a full classification of rank 1 solutions. We have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial (zero) solution. The wave fronts are non-stationary and they propagate with a constant average velocity. The system also has soliton solutions similar to breathers, which resembles soliton webs in the KP theory. We associate the classification of soliton solutions with the Schubert decomposition of the Grassmannians Gr R ( k , N ) and Gr C ( k , N ) .

      PubDate: 2017-02-09T06:01:16Z
      DOI: 10.1016/j.physd.2017.01.003
  • The stability spectrum for elliptic solutions to the focusing NLS equation
    • Authors: Bernard Deconinck; Benjamin L. Segal
      Abstract: Publication date: Available online 23 January 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Bernard Deconinck, Benjamin L. Segal
      We present an analysis of the stability spectrum of all stationary elliptic-type solutions to the focusing Nonlinear Schrödinger equation (NLS). An analytical expression for the spectrum is given. From this expression, various quantitative and qualitative results about the spectrum are derived. Specifically, the solution parameter space is shown to be split into four regions of distinct qualitative behavior of the spectrum. Additional results on the stability of solutions with respect to perturbations of an integer multiple of the period are given, as well as a procedure for approximating the greatest real part of the spectrum.

      PubDate: 2017-01-28T05:15:28Z
      DOI: 10.1016/j.physd.2017.01.004
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